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Geodesics in Lorentzian Manifolds California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 3-2016 GEODESICS IN LORENTZIAN MANIFOLDS Amir A. Botros California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Geometry and Topology Commons Recommended Citation Botros, Amir A., "GEODESICS IN LORENTZIAN MANIFOLDS" (2016). Electronic Theses, Projects, and Dissertations. 275. https://scholarworks.lib.csusb.edu/etd/275 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Geodesics in Lorentzian Manifolds A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Amir Amin Botros March 2016 Geodesics in Lorentzian Manifolds A Thesis Presented to the Faculty of California State University, San Bernardino by Amir Amin Botros March 2016 Approved by: Dr. Corey Dunn, Committee Chair Date Dr. Wenxiang Wang, Committee Member Dr. Rolland Trapp, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Mani- folds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between \events". They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called \geodesically complete", or simply, \complete". It is easy to show that g(σ0; σ0) is constant if σ is a geodesic and g is a smooth metric on a manifold M, so one can characterize geodesics in terms of their causal character: if g(σ0; σ0) < 0, σ is timelike. If g(σ0; σ0) > 0, σ is spacelike. If g(σ0; σ0) = 0, then σ is lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the notions of spacelike complete, timelike complete, and null or lightlike complete. We illustrate that some of the notions are inequivalent. iv Acknowledgements It is an honor and my most heartfelt thanks to acknowledge Dr. Dunn because I have with him useful conversations on many matters. Dr. Dunn offered invaluable assistance by reading my manuscript carefully and by providing helpful comments, both stylistic and mathematical. I am indebted to Dr. Dunn, who has made helpful suggestions concerning the revisions of this thesis. My thoughtful input to Dr. Dunn, who served as viewer for each detail of this thesis. It is sincerely appreciated and grateful to Dr. Dunn because Dr. Dunn put too much time working on this thesis to become as it is now. It is my pleasure to say that Dr. Dunn's tongue is the pen of ready writer to me. It is an honor and my most heartfelt thanks to acknowledge Dr. Trapp, the first Professor taught me Mathematics in California State University San Bernardino and also in USA. Dr. trapp taught me a lot of rules that how to think very fast to solve problems. I am indebted to Dr. Trapp because Dr. Trapp always gave me a better grade than I did in his class. It is an honor and my most heartfelt thanks to acknowledge Dr. Wang who make me to like manifolds. My thoughtful input to Dr. Wang because when I asked him to take a copy from his notes about my math class that he taught me, he always gave to me. Finally, I thank all of my committee because they accepted me as a student and allow me to research in this thesis. v Table of Contents Abstract iii Acknowledgements iv 1 Introduction 1 2 Topological and Differential Geometric Preliminaries 3 3 Inner Products and Pseudo Riemannian Manifolds 8 4 Connection and Geodesics Completeness with Examples in Timelike, Spacelike, and Lightlike 13 4.1 Connection . 13 4.2 Geodesics and Completeness . 22 4.3 Summary . 31 Bibliography 33 1 Chapter 1 Introduction A geodesic on a Riemannian manifold is, locally, a length minimizing curve. In other words, a geodesic is a path that a non-accelerating particle would follow. For example, a geodesic in the Euclidean plane is a straight line and on the sphere, all 2 P4 2 geodesics are great circles. If we consider the distance function jjXjj = i=1 Xi = 2 2 2 2 4 t + x + y + z on R , we notice that it is positive definite (Riemannian). Moreover, the Hopf-Rinow Theorem states that a connected Riemannian manifold is geodesically complete if and only if it is complete as a metric space [Lee97]. In 1915 Albert Einstein introduced the general theory of relativity, which led to the need to consider manifolds whose metric is not positive definite (pseudo-Riemannian). In other words, pseudo-Riemannian manifolds are not metric spaces, since the distance 2 2 P4 2 2 2 2 2 function, jjXjj = −X1 + 2 Xi = −t + x + y + z is no longer positive definite and geodesics here can be viewed as a distance between \events". They are no longer distance minimizing (instead, some are distance maximizing) or zero. There is no analog of the Hopf-Rinow Theorem in the pseudo-Riemannian case. So, completeness is defined as geodesic completeness. A manifold is geodesically complete if every geodesic extends for infinite time and here are 3 types of geodesic completeness. 1. Timelike geodesically complete if every timelike geodesic extends for infinite time. 2. Spacelike geodesically complete if every spacelike geodesic extends for infinite time. 3. Lightlike geodesically complete if every lightlike geodesic extends for infinite time. It is our goal to show that these notions are inequivalent. 2 Here is an outline of the thesis. Chapter 2 contains Topological and Differential Geometric Preliminaries. We introduce topological spaces, smooth manifolds, and related notions that we will need. Chapter 3 is about Inner Products and pseudo-Riemannian manifolds. We in- troduce inner products, timelike, lightlike, spacelike vectors, and pseudo-Riemannian manifolds. Chapter 4 is about Connections and Geodesics Completeness with examples that demonstrate certainty of completeness are inequivalent. We introduce connections, Levi- Civita connections, Koszul formula, Geodesics Completeness, special manifold, examples, and finally, give a short summary of our goal. 3 Chapter 2 Topological and Differential Geometric Preliminaries . Most of this material that I introduce here in this chapter can be found in a basic topology book. The concept of topological space grew out of the study of the real line and Euclidean space and the study of continuous functions on these spaces ([Mun00]). In this chapter we define what a topological space is, and we provide some examples. Definition 2.1. A topological space is a set X together with a collection of subsets S, called open sets, that satisfies the four conditions: 1. The empty set ; is in S. 2. X is in S. 3. The intersection of a finite number of sets in S is also in S. 4. The union of an arbitrary number of sets in S is also in S. Here we gave a basic example of topological space over a set to give some of the elementary concepts to understand a topological space. Example 2.2. If X = f1; 2; 3g then S = fφ, f1g ;Xg is a topology on X. 4 A Euclidean space or, more precisely, a Euclidean n−space, is the generalization of the notions \plane" and \space" (from elementary geometry) to arbitrary dimensions n. Thus Euclidean 2−space is the plane, and Euclidean 3−space is space. This generalization is obtained by extending the axioms of Euclidean geometry to allow n directions which are mutually perpendicular to each other. Euclidean n−space, sometimes called Cartesian space or simply n−space, is the space of all n−tuples of real numbers, (x1; x2; :::; xn). Such n−tuples are sometimes called points. n Definition 2.3. Euclidean n−space R is the set of all n−tuples p = (p1; p2; : : : ; pn) of a real numbers. We work here over the real numbers, R. n Definition 2.4. For any x in R and r > 0 , the open ball of radius r around x is subset Br(x) = fy 2 R; such that j x − y j< rg. n In a moment we will give an example of the standard topology on R . n Example 2.5. If X = R then let S be the set of arbitrary unions of open balls. This is n the standard topology on R we will be using. In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain. n Definition 2.6. A real valued function f defined on an open set U of R is smooth if all mixed partial derivatives of f and of all orders exist and are continuous at every point of U. n m If U ⊆ R and f : U ! R is smooth then f = ff1; f2; : : : ; fmg where fi :! R. These fi; i = 1; 2; :::::; m, are called the coordinate functions of f, and f is smooth if all of the coordinate functions of f are smooth [Shi04].
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